Consider the problem of a monopolist that sells its product on two different markets m, with m=1,2. Each market has an aggregate demand function given by 1200−α_m*p_m, where p_m denotes the price in market m, and α_m=m measures the responsivity of demand to prices in market m.
The monopolist's cost function is given by c(q)=12q2, where q denotes the total amount produced for all markets.
The monopolist is owned by a foreign company, so none of the monopolist's profits are received by the consumers in these markets.
The law allows the monopolist to charge different pricees in different markets, but does not allow any other forms of price discrimination.
(i)What is the equilibrium level of production in market 2?
(ii)What is total consumer surplus in the economy (i.e., taking both markets into account)?
(iii)Suppose that the government behind market 1 introduces a tax of $100 per unit on the monopolist's sales in its market (paid by the firm), and that the tax revenue is given back to consumers in market 1 using lump-sum transfers. Suppose also that no such tax is introduced by the government behind market 2.
What is the new equilibrium level of production in market 2?
(iv)What is the the new total level of consumer surplus in the economy (including the tax revenues)?
To solve this problem, we need to find the equilibrium level of production in market 2, the total consumer surplus, the new equilibrium level of production in market 2 after the introduction of a tax, and the new total level of consumer surplus in the economy.
(i) Equilibrium Level of Production in Market 2:
To find the equilibrium level of production in market 2, we need to find the point where the monopolist's marginal cost equals the marginal revenue corresponding to market 2's demand function.
The monopolist's cost function is given by c(q) = 12q^2, where q denotes the total amount produced for all markets.
The marginal cost (MC) is the derivative of the cost function with respect to quantity, c'(q) = 24q.
The marginal revenue (MR) corresponding to market 2's demand function is the derivative of the demand function with respect to price, MR = α_2(1200 - α_2*p_2)'.
Differentiating the demand function with respect to price, we get (1200 - α_2*p_2)' = -α_2.
Setting MR equal to MC, we have α_2(1200 - α_2*p_2)' = MC,
α_2(1200 - α_2*p_2)' = 24q.
Solving this equation, we can find the equilibrium level of production in market 2.
(ii) Total Consumer Surplus in the Economy:
To find the total consumer surplus in the economy, we need to calculate the area under the demand curve for both markets.
The consumer surplus for market 1 is given by the integral of the demand function, CS_1 = ∫(0 to q_1) (1200 - α_1*p_1) dp_1, where q_1 is the equilibrium level of production in market 1.
Similarly, the consumer surplus for market 2 is given by CS_2 = ∫(0 to q_2) (1200 - α_2*p_2) dp_2, where q_2 is the equilibrium level of production in market 2.
The total consumer surplus in the economy is the sum of CS_1 and CS_2.
(iii) New Equilibrium Level of Production in Market 2:
After the introduction of a tax in market 1, the monopolist's revenue in market 1 will decrease by the amount of the tax per unit sold.
The new demand function in market 1 becomes 1200 - α_1*(p_1 + tax), where p_1 is the price in market 1.
To find the new equilibrium level of production in market 2, we follow the same steps as in part (i) but use the new demand function.
(iv) New Total Level of Consumer Surplus in the Economy:
To find the new total level of consumer surplus in the economy after the introduction of the tax, we calculate the area under the new demand curve for both markets using the new equilibrium levels of production.
The consumer surplus in market 1 will be affected by the tax, so we need to calculate the area under the new demand curve, CS_1_new = ∫(0 to q_1_new) (1200 - α_1*(p_1 + tax)) dp_1.
The consumer surplus in market 2 remains the same as in part (ii), CS_2.
The new total consumer surplus in the economy is the sum of CS_1_new and CS_2, plus the tax revenue generated.
Remember to substitute the values of α_1, α_2, and the tax amount into the equations before calculating the integrals.
Using these steps, you can solve this problem and find the answers to each part.